A number of older papers by V.P. Platonov (in Russian, often followed by English translations) deal with periodic linear groups or linear algebraic groups in which the notions of Sylow theory make sense and where some results from the finite case actually generalize. One of the more substantial papers deals especially with conjugacy theorems:

In other papers Platonov also works with classes of topological groups in a similar spirit.

P.S. Concerning sources, the long 1966 paper appears in an English translation (by the group theorist Kurt Hirsch) in volume 69 of the AMS Translations (Series 2), 1969; but this doesn't seem to be accessible online. There is a Google Scholar entry containing a full text PDF version of the Russian original here.

Amalgams of finite groups provide another example. Let $A$ and $B$ be finite groups and let $C = A \cap B.$ Suppose that $P$ is a Sylow $p$-subgroup of $A$, and that $C$ contains a Sylow $p$-subgroup of $B.$ Then the amalgam $A*_{C}B$ has a unique conjugacy class of maximal finite $p$-subgroups, but is an infinite group as long as $C$ is proper in both $A$ and $B.$. In fact, the process an then iterated to the case where $A$ and $B$ may themselves be amalgams of finite groups of this type, and so on. For general results on amalgams, see J-P. Serre's book "Trees". For applications of this type of construction to fusion systems on finite $p$-groups, see two recent papers of mine in Journal of Algebra and Transactions of the AMS.